Suppose that a finite set of polynomials in C[x,y,z] has a finite number of solutions (i.e. the generated ideal is 0-dimensional).
Suppose also that the Groebner basis wrt plex(x,y,z) is
[f(z), g(y,z), h(y,z), k(x,y,z)]
As well known, the system can be now easily solved: choose a root z0 of f, plug it into g and h and look for a common root (y0) etc.
The question is the following:
Is it true that for EVERY root z0 of f there exist y0, z0 such that (x0,y0,z0) satisfy the system?
In all the examples I have seen this is true, but I don't know whether this is true in general or there is a counterexample.
[This is not a pure Maple question but I know that some members here work in this area].